On the wave turbulence theory of 2D gravity waves, II: propagation of randomness

This is the second part of our work initiating the rigorous study of wave turbulence for water waves equations. We combine energy estimates, normal forms, and probabilistic and combinatorial arguments to complete the construction of long-time solutions with random initial data for the 2d (1d interface) gravity water waves system on large tori. This is the first long-time regularity result for solutions of water waves systems with large energy (but small local energy), which is the correct setup for applications to wave turbulence. Such a result is only possible in the presence of randomness.